The Fourier transform of a function ƒ(t) has served as the most important transform in numerous signal processing applications. For example, the Fourier transform is widely used in imaging analysis such as CT and Magnetic Resonance Imaging (MRI).
Standard Fourier analysis reveals individual frequency components involved in a signal or image. However, in many situations of frequencies changing over time or space the standard Fourier analysis does not provide sufficient information. In numerous applications processing of non-stationary signals or images reveals important information. For example, in MRI signal processing motion caused by respiratory activity, cardiac activity, blood flow causes temporal changes in spatial frequencies.
To overcome the deficiency of the standard Fourier analysis, other techniques such as the Gabor transform (GT) disclosed in: Gabor, D. “Theory of communications”, J. Inst. Elec. Eng., 1946; 93, 429-457, also known as the short time Fourier transform, and the Wavelet transform (WT) disclosed in: Goupillaud P., Grossmann, A., Morlet J. “Cycle-octave and related transforms in seismic signal analysis”, Geoexplor, 1984; 23, 85-102, and in: Grossmann, A., Morlet J. “Decomposition of Hardy functions into square integrable Wavelets of constant shape”, SIAM J. Math. Anal., 1984; 15, 723-736, have been developed, references to which are incorporated herein by reference. Both of these methods unfold the time information by localizing the signal in time and calculating its “instantaneous frequencies.” However, both the GT and the WT have limitations substantially reducing their usefulness in the analysis of imaging signal data. The GT has a constant resolution over the entire time-frequency domain which limits the detection of relatively small frequency changes. The WT has variant resolutions, but it provides time vs. scale information as opposed to time vs. frequency information. Although “scale” is loosely related to “frequency”—low scale corresponds to high frequency and high scale to low frequency—for most wavelets there is no explicit relationship between scale factors and the Fourier frequencies. Therefore, the time-scale representation of a signal is difficult if not impossible to interpret.
In 1996 another approach for time-frequency analysis has been disclosed in 1D form by Stockwell R. G., Mansinha L., Lowe R. P., “Localization of the complex spectrum: the S-transform”, IEEE Trans. Signal Process, 1996; 44, 998-1001. In 1997 the two-dimensional S transform has been disclosed in Mansinha L., Stockwell R. G., Lowe R. P., “Pattern analysis with two dimensional localization: Applications of the two dimensional S transform”, Physica A., 1997; 239, 286-295, and in Mansinha L., Stockwell R. G., Lowe R. P., Eramian M., Schincariol R. A., “Local S spectrum analysis of 1D and 2D data”, Phys. Earth Planetary Interiors, 1997; vol. 239, no. 3-4, 286-295, references to which are incorporated herein by reference. The ST has been successfully applied in analyzing temporal or spatial variances of the spectrum of a time series or an image, respectively.
However, the ST of a 2D image function I(x, y) retains the spectral variables kx and ky as well as the spatial variables x and y, resulting in a complex-valued function of four variables. Therefore, visualization of 2D ST results is a difficult task requiring substantially large computer storage capability and long computer processing time.
There have been several different approaches for visualizing the 2D ST disclosed in the recent literature. In Mansinha L., Stockwell R. G., Lowe R. P., “Pattern analysis with two dimensional localization: Applications of the two dimensional S transform”, Physica A., 1997; 239, 286-295, slicing of the S space along x or y directions has been disclosed. Such a method provides an understanding of local information but does not supply a continuous picture of frequency variation over space.
Dominant wave number mapping, disclosed in Mansinha L., Stockwell R. G., Lowe R. P., Eramian M., Schincariol R. A., “Local S spectrum analysis of 1D and 2D data”, Phys. Earth Planetary Interiors, 1997; vol. 239, no. 3-4, 286-295, records the maximum magnitude in the local spectrum at every spatial point. If the frequency component with the maximum magnitude dominates the local spectra the local dominant wave number map reveals distinctive subregions or textures. However, when more than one dominant frequency component appears in the local spectral domain choosing one with maximum amplitude results in a loss of important details in the structure.
In Eramian M. G., Schincariol R. A., Mansinha L., and Stockwell R. G., “Generation of aquifer heterogeneity maps using two-dimensional spectral texture segmentation techniques”, Math. Geol., 1999, vol. 31, no. 3, 327-348, the binary encoded S transform (BEST) method representing each local spectrum with a binary number is disclosed. However, due to non-linear weighting of the local spectrum, slight changes at high frequencies cause very different values in corresponding binary number assignments. Furthermore, ordering of the subregions is not unique, resulting in different interpretations of the texture. Thus the BEST method provides reliable results only when the dominant frequency components occur where the power of the spectrum is evenly distributed.
Eramian et al. further disclosed in the above reference the angular difference method ADM. The ADM provides a relative angular difference between a local spectrum and a reference spectrum allowing measuring of spectral variations with respect to the reference spectrum. However, the ADM is insensitive to subtle changes at frequencies not present in the reference spectrum.
In Oldenburger G. A., “Characterizing the spatial distribution of hydraulic conductivity: Application of ground penetrating radar and space-local spectral techniques”, M.Sc. Thesis, University of Western Ontario, London, 2000, and Oldenburger G. A., Schincariol R. A., and Mansinha L. “Space-local spectral texture segmentation applied to characterizing the heterogeneity of hydraulic conductivity”, Water Resources Research, 38, 10.1029/2001 WR000496 (2002) a combination of the ADM and BEST methods using a weighted scalar product (WSP) with a reference matrix is disclosed. Here, a good choice of the reference matrix is crucial since it determines how different frequency components are distinguished.
Summarizing the above analysis of the prior art, all the above methods require prior knowledge about frequency content in an image for visualizing ST data.